Binary Strings

INFO

A sequence composed solely of the characters 0 and 1.
These strings are fundamental in computer science and digital systems—used to represent binary data, machine instructions, and logic states.

How Many Binary Strings of Length $n$?

$$|\{0,1\}\times \{0,1\}\times \cdots \times \{0,1\}|=|\{0,1\}{|}^n=2^n$$

IMPORTANT

In general, the number of strings of length $n$ over an alphabet with $X$ characters is:

$$X^n$$

• $X$: number of choices per position
• $n$: number of positions

See also: Power Rule

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Examples

1. Latin Alphabet Words

How many 4-letter words can be formed over the Latin alphabet (26 letters)?

$$26^4$$

2. Passwords with Mixed Characters

How many 8-character passwords can be made using uppercase, lowercase, and digits?

$$(26+26+10{)}^8=62^8$$

3. Passwords with Disjoint Character Sets

How many 8-character passwords contain only uppercase, only lowercase, or only digits?

$$26^8+26^8+10^8$$

NOTE

This is a direct application of the Sum Rule.

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Complement Method: “At Least One” Conditions

4. At Least One Uppercase Letter

How many 8-character passwords contain at least one uppercase letter?

Let:

• Total passwords: $62^8$
• Passwords with no uppercase (only lowercase + digits): $36^8$

Then:

$$\text{At least one uppercase}=62^8-36^8$$

Alternate form:

$$26\cdot 62^7$$

TIP

This is a classic use of the complement principle. See also: Inclusion Exclusion

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Complement Method: “At Least One 0”

5. 4-Digit Strings with at Least One 0

How many 4-digit strings (digits 0–9) contain at least one 0?

• Total: $10^4$
• No 0s (digits 1–9 only): $9^4$

So:

$$10^4-9^4$$

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Exact Count Methods

6. Exactly $k$ Zeros in a 4-Digit String

Breakdown:

$$(4\cdot 9^3)+(6\cdot 9^2)+(4\cdot 9)+1$$

7. First Occurrence of 0

Alternative breakdown:

$$10^3+(9\cdot 10^2)+(9^2\cdot 10)+9^3$$

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Distribution Interpretation

8. Distributing 10 Distinct Candies to 4 Children

If each candy independently goes to one of 4 children:

$$4^{10}$$

If each child gets exactly one candy and order matters:

$$10\cdot 9\cdot 8\cdot 7=P(10,4)$$

NOTE

See Power Rule and rPermutations for context.